Parabolic conjugation on nilpotent elements for classical Lie types

设G是单经典复李群，g是它的李代数.设N是g中幂零元的幂零锥，用N（m）表示幂零度为m的幂零元n的子簇，即n^m=0。考虑G的任意标准抛物子群在簇N（m）上的共轭作用。我们的主要目的是证明一个标准，列出所有的情况下，抛物子群和整数m所描述的行动只有100多个轨道。我们希望详细了解有限的情况下，例如在参数化的轨道组合对象，通过描述退化的轨道和计算奇异点的轨道封闭。在无限的情况下，我们打算描述无限家庭的轨道，并希望定义半不变量，产生的抛物半不变rings.By翻译的语言中的有限维代数的关系，通过箭图的设置到一个表示理论的上下文中，许多所描述的问题已经回答的情况下，G是一般的线性群。一个类似的翻译是可能的辛和正交李类型-在这里，对称箭图与关系的对称表示必须考虑，这使得有可能证明第一结果的情况下，m=2。我们的目标是使用这个翻译，以证明我们所期望的结果，扩展到目前为止已知的对称表示理论。G是一般线性群的情况给出了许多关于如何在这里进行的线索。

Let G be a simple classical complex Lie group and let g be its Lie algebra. Let N be the nilpotent cone of nilpotent elements in g and denote by N(m) the subvariety of such nilpotent elements n of nilpotency degree m, that is, n^m=0. We consider the conjugation-action of an arbitrary standard parabolic subgroup of G on the variety N(m). Our main aim is to prove a criterion which lists all cases of parabolic subgroups and integers m for which the described action only has finitely many orbits. We want to understand the finite cases in detail, for example in terms of a parametrization of the orbits by combinatorial objects, by describing degenerations of orbits and by calculating singularities in orbit closures. In the infinite cases, we intend to describe infinite families of orbits and would like to define semi-invariants which generate the parabolic semi-invariant rings.By translating the setup to a representation-theoretic context in the language of finite-dimensional algebras by means of quivers with relations, many of the described questions have been answered in the case of G being the general linear group. A similar translation is possible for symplectic and orthogonal Lie types - here, symmetric representations of symmetric quivers with relations have to be considered, which made it possible to prove first results for the case that m=2. We aim to use this translation in order to prove our aspired results by expanding the so far known symmetric representation theory. The case where G is the general linear group gives many clues on how to proceed here.